Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights
-
Javier Martínez Perales
Abstract
In this paper, we review certain extensions of the Gagliardo and Poincaré- Sobolev-type inequalities to later explore the possibility of extending them to the vectorvalued setting. We restrict ourselves to the most classical case of ℓq-valued functions, where already some difficulties arise, due to the lack of a vector-valued variant of the truncation method, both on the classical and the fractional case. We think that these difficulties may be overcome in the future, and we pose some conjectures in this direction.
Abstract
In this paper, we review certain extensions of the Gagliardo and Poincaré- Sobolev-type inequalities to later explore the possibility of extending them to the vectorvalued setting. We restrict ourselves to the most classical case of ℓq-valued functions, where already some difficulties arise, due to the lack of a vector-valued variant of the truncation method, both on the classical and the fractional case. We think that these difficulties may be overcome in the future, and we pose some conjectures in this direction.
Chapters in this book
- Frontmatter I
- Foreword V
- Contents VII
- Yau’s conjecture on the dimension of harmonic polynomials 1
- On Carrasco Piaggio’s theorem characterizing quasisymmetric maps from compact doubling spaces to Ahlfors regular spaces 23
- On trace theorems for weighted mixed-norm Sobolev spaces and applications 49
- Sharp stability of the logarithmic Sobolev inequality in the critical point setting 77
- On a variational problem of nematic liquid crystal droplets 103
- Estimates for the variable order Riesz potential with applications 127
- A remark on the atomic decomposition in Hardy spaces based on the convexification of ball Banach spaces 157
- A Bliss–Adams inequality 179
- Trudinger-type inequalities in RN with radial increasing mass-weight 197
- In response to David R. Adams’ October 12, 2001, letter 215
- Some remarks on capacitary integrals and measure theory 235
- Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights 265
- Index 287
Chapters in this book
- Frontmatter I
- Foreword V
- Contents VII
- Yau’s conjecture on the dimension of harmonic polynomials 1
- On Carrasco Piaggio’s theorem characterizing quasisymmetric maps from compact doubling spaces to Ahlfors regular spaces 23
- On trace theorems for weighted mixed-norm Sobolev spaces and applications 49
- Sharp stability of the logarithmic Sobolev inequality in the critical point setting 77
- On a variational problem of nematic liquid crystal droplets 103
- Estimates for the variable order Riesz potential with applications 127
- A remark on the atomic decomposition in Hardy spaces based on the convexification of ball Banach spaces 157
- A Bliss–Adams inequality 179
- Trudinger-type inequalities in RN with radial increasing mass-weight 197
- In response to David R. Adams’ October 12, 2001, letter 215
- Some remarks on capacitary integrals and measure theory 235
- Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights 265
- Index 287
